Extremal Combinatorics VI: The Frankl-Wilson Theorem

The Frankl-Wilson theorem is a remarkable theorem with many amazing applications. It has several proofs, all based on linear algebra methods (also referred to as dimension arguments). The original proof is based on a careful study of incidence matrices for families of sets. Another proof by Alon, Babai and Suzuki applies the “polynomial method”. I will describe here one variant of the theorem that we will use in connection with the Borsuk Conjecture (It follows a paper by A. Nilli). I will post separately about more general versions, the origin of the result, and various applications. One application about the maximum volume of spherical sets not containing a pair of orthogonal vectors is presented in the next post.

Let and suppose that is an odd prime.

Theorem: Let be a subset of with the property that no two vectors in are orthogonal. Then .

We will prove a slightly modified version which easily implies the Theorem as we stated it before:

Let be the set of vectors () in such that and the number of ‘1’ coordinates is even. Let be a subset of with the property that no two vectors in are orthogonal. Then .

Proof:

The ad-hoc trick:

Claim 1: Let . If then .

This is a crucial part of the argument and as far as I can see the part that you need to replace entirely in other applications of the method.

We need

Claim 1′: For every , .

Given Claim 1′, if then . Since the first coordinate is 1 it is not possible that and it follows that .

Proof (of Claim1′): Suppose that the number of coordinates where and is , the number of coordineates where and is , the number of coordineates where and is , and the number of coordineates where and is . The inner product . Now is even and also is even so is also even and is divisible by 4. is divisible by 4 and therefore so is .

The algebra-puzzle trick

Compute .

The answer is ZERO because one of the terms is (x-x).

Magic: underline the empty line with your mouse to see the solution.

Define a family of polynomials in variables , over the field with elements as follows. For every define . Here .

Claim 2: For , we have .

Proof: Since we must have that for some , . Therefore one of the terms in the product defining must vanish. (Remember, we work over the field with elements. Ahla!

Claim 3: .

Proof: The inner product of a vector with itself in . This is important! Therefore . (It is a product of non-zero numbers modulo . What is modulo p? This is a result of a different Wilson.) Walla!

Remark: Some people prefer to simply write which gives a different less extravagant way to present the same thing. (And without the algebra-puzzle.) I still prefer the way we defined the polynomials because for other applications of the method, it generalizes more easily.

The linear algebra (diagonal) trick

Claim 4: The polynomials are linearly independent.

Proof: Suppose that . Substitute in this sum for . By Claim 2 we obtain that the outcome is . By claim 3 and therefore . Sababa!

Remark: It is worthwhile to remember a variation of this trick, the linear algebra upper-diagonal trick, where you conclude linear independence when you assume that there is some order on your set of ‘s and that for .

The multilinearization trick

We now replace the polynomials by new polynomials by applying the following rule:

Replace each by 1 and each by !

Claim 5: The new polynomials are square-free, and they attain the same values for vectors with coordinates .

Proof: This is because when we have a variable that attains only the values +1 and -1 the value of is 1, and the value of is the same as that of . Walla!

Claim 6: The polynomials are linearly independent.

Proof: The proof of claim 4 applies. Walla!

The end of the proof:

By claim 6, the size of is at most the dimension of the vector space of square-free polynomials of degree in variables . This dimension is the number of sqauare-free monomials of degree in variables which is .

14 thoughts on “Extremal Combinatorics VI: The Frankl-Wilson Theorem”

The reason the second version implies the first is as follows: If we flipp the value of the first coordinate we do not change inner products so the second version applies also to subsets of which is the set of all vectors with having odd numbers of ‘1’s. The proof only used the fact that the number of 1 coordinates is even and there are no antipodal vectors . Therefore, we can simply cover all vectors by 4 sets to which the second version applies.

Thanks! 1) If you look at the set of all the vectors with at most ‘-1’s or at most +1’s you get an example with no two orthogonal vectors, with about half the number in the theorem. 2) If n is even but not of the form we considered you can apply Frankl-Rodl’s theorem to get bounds of the form with worth value of . (The Frankl Wilson theorem gives the correct value of in the four times prime case.)

Dear Prof. Gil Karai,
I really don’t understand your argument at the end of the proof : “number of square-free monomials of degree p-1 in n variables which is .”
For example, , square-free monomials of degree 2 in 12 variables are , hence there are 13 square-free monomials in 12 variables. BUT your answer is . Could you please explain?
Many thanks,